Abstract
Unconventional superconductivity in molecular conductors is observed at the border of metalinsulator transitions in correlated electrons under the influence of geometrical frustration. The symmetry as well as the mechanism of the superconductivity (SC) is highly controversial. To address this issue, we theoretically explore the electronic properties of carrierdoped molecular Mott system κ(BEDTTTF)_{2}X. We find significant electronhole doping asymmetry in the phase diagram where antiferromagnetic (AF) spin order, different patterns of charge order, and SC compete with each other. Holedoping stabilizes AF phase and promotes SC with d_{xy}wave symmetry, which has similarities with highT_{c} cuprates. In contrast, in the electrondoped side, geometrical frustration destabilizes the AF phase and the enhanced charge correlation induces another SC with extendeds + \(d_{x^2  y^2}\)wave symmetry. Our results disclose the mechanism of each phase appearing in fillingcontrol molecular Mott systems, and elucidate how physics of different stronglycorrelated electrons are connected, namely, molecular conductors and highT_{c} cuprates.
Introduction
Understanding the intimate correlation among metalinsulator (MI) transition, magnetism, and superconductivity (SC) is one of the most challenging issues in modern condensed matter physics. The most wellstudied example is the highT_{c} cuprates, where SC is observed when mobile carriers are doped into the parent antiferromagnetic (AF) Mott insulators^{1,2,3}. A general understanding there, supported by various experiments and theories, is that strong AF spin fluctuation mediates the dwave SC that appears through the fillingcontrol Mott MI transition generating mobile charge carriers. However, can we export this mechanism to other strongly correlated materials? To address this question, it is crucial to make a comparison among different classes of materials. In this respect, heavy fermion compounds and molecular conductors provide such opportunities^{4,5,6,7,8}.
The family of quasi twodimensional molecular conductors κ(ET)_{2}X (ET = BEDTTTF, and X takes different monovalent anions^{9}) is in fact compared often with the cuprates^{10}. They indeed have common factors: simple quasitwodimensional electronic structure to begin with in the noninteracting limit and the Mott MI transition and SC closely related with each other. However, there are important differences: First, in κ(ET)_{2}X, SC appears through the bandwidthcontrol Mott transition; the carrier density is usually unchanged but the pressure (either physically or chemically) is the controlling factor. Although the variation of the carrier density is necessary for direct comparisons, it has not been realized in κ(ET)_{2}X for a long time due to experimental difficulties. Second, while the cuprates are basically governed by the physics nearby 1/2filling, κ(ET)_{2}X is a 3/4filled system. The similarity enters when the socalled dimer approximation is applied in the latter^{11}, resulting in the effective 1/2filled system (dimer model). Although the dimer model has been extensively studied using various theoretical methods^{12,13,14,15,16,17,18,19,20,21,22,23,24,25}, the validity of the dimer approximation itself is recently reexamined^{26,27,28,29,30,31,32}. Especially, the importance of the intradimer charge degree of freedom^{33,34} and intersite Coulomb interactions^{27,29}, which are discarded in the dimer approximation, has attracted much attention because of recent experimental suggestions^{35,36,37,38}. Third, in κ(ET)_{2}X, SC is observed not only next to the AF insulators but also to nonmagnetic (candidate of gapless spinliquid) insulators^{8,39,40,41}. The strong influence of geometrical frustration owing to the anisotropic triangular arrangement of dimers is present in this family.
Recently, carrier doping has been realized either chemically in κ(ET)_{4}Hg_{3−δ}Y_{8} (Y = Br or Cl)^{42,43,44,45} or in κ(ET)_{2}Cu[N(CN)_{2}]Cl (κCl) by using electricdoublelayer transistor (EDLT) technique^{46,47}, revealing intriguing phenomena such as a domeshaped SC region, anomalous metallic behaviors, and significant electronhole doping asymmetry, which are all reminiscent of the highT_{c} cuprates. Therefore, κtype ET systems can now provide a unique playground of both fillingcontrol and bandwidthcontrol Mott transitions with SC phases nearby, for which a unified theoretical understanding is highly desired.
In this paper, we theoretically study the groundstate properties of κ(ET)_{2}X varying the carrier number from 3/4filling, in order to elucidate the electronic phases appearing near the Mott transition in this system, especially SC, and to investigate their stabilities beyond mean field treatments. The intradimer charge degree of freedom and intersite Coulomb interactions are explicitly considered. We find that the groundstate phase diagram shows significant electronhole asymmetry in the stability of AF phase and in terms of competing two types of SC. While in the holedoped side d_{xy}wave SC is favored by the AF spin fluctuation as in the highT_{c} cuprates, the electron doping highlights the geometrical frustration and the charge degree of freedom, which are unique in κ(ET)_{2}X, stabilizing extendeds +\(d_{x^2  y^2}\)wave SC. The electronhole doping asymmetry, including the symmetry of SC, is attributed to the degree of frustration that is controlled by carrier doping. This is a conceptually new perspective to κ(ET)_{2}X, which can also be applied to other frustrated systems in general. Our results, beyond the usual description based on the 1/2filled dimer model, thus provide new understanding of how physics of molecular conductors and highT_{c} cuprates are distinct.
Results
Model derivation and framework
The electronic properties of molecular conductors are modeled by a simple model where the molecules are replaced by lattice sites^{48}. They are described by the extended Hubbard model (EHM)^{11,49}, a textbook model for studying correlated electrons. The Hamiltonian is given as
where \(c_{i\sigma }^\dagger\) (c_{iσ}) is a creation (annihilation) operator of electron at molecular site i with spin σ(=↑,↓), \(n_{i\sigma } = c_{i\sigma }^\dagger c_{i\sigma }\), and n_{i} = n_{i↑} + n_{i↓}. U and V_{ij} are onsite and intersite Coulomb repulsions, respectively. <i, j> denotes pairs of neighboring molecules in the κtype geometry, labeled by b_{1}, b_{2}, p, and q, as shown in Fig. 1a.
The tightbinding parameters t_{ij} are set for the deuterated κ(ET)_{2}Cu[N(CN)_{2}]Br (κBr), which locates very close to the MI transition^{50}, and are adopted from a firstprinciples band calculation as \((t_{b_1},t_{b_2},t_p,t_q) = (196,65,105,  39)\,{\mathrm{meV}} = (1.0,0.332,0.536,  0.199)\,t_{b_1}\)^{51}. We set the largest hopping integral \(t_{b_1}\) as the unit of energy. The unit cell is a rectangle with R_{x} × 2R_{y} and δ_{±} = (δ_{x}, ±δ_{y}) are vectors connecting the centers of molecules facing each other in a dimer (see Fig. 1a). Here, we set (R_{x}, R_{y}, δ_{x}, δ_{y}) = (1.0, 0.7, 0.3, 0.3)R_{x} with R_{x} as a unit of length^{29}. The noninteracting band structure is shown in Fig. 1b. Among the four energy bands, the upper two bands (bands 1 and 2) contribute to form the Fermi surface (FS). For the undoped case, the electron density per molecular site is n = 3/2 (3/4filling) and it corresponds to the hole density n_{hole} = 2 − n = 1/2. In this study, we change n_{hole} from 1/3 to 2/3 to investigate the doping dependence of the system. The corresponding Fermi energies are indicated in Fig. 1b by dotted lines.
The effect of Coulomb interactions is treated using a variational Monte Carlo (VMC) method^{52,53,54}. The trial wave function considered here is a Gutzwiller–Jastrow type, \(\left \Psi \right\rangle = P_{{\mathrm{J}}_{\mathrm{c}}}P_{{\mathrm{J}}_{\mathrm{s}}}\left {\mathrm{\Phi }} \right\rangle\). Φ> is a onebody part constructed by diagonalizing the onebody Hamiltonian, and \(P_{{\mathrm{J}}_{\mathrm{c}}}\) and \(P_{{\mathrm{J}}_{\mathrm{s}}}\) are charge and spin Jastrow factors, respectively. The explicit form of them are described in Methods. In the following, we show results for 1152 molecular sites (corresponding to L = 24, see in Methods), which is large enough to avoid finite size effects.
Groundstate phase diagram
Figure 2a shows the groundstate phase diagram. The hole density n_{hole} = 2 − n and the onsite Coulomb interaction \(U{\mathrm{/}}t_{b_1}\) are varied as parameters, while the ratio between U and the largest intersite Coulomb interaction \(V_{b_1}\) is fixed at \(V_{b_1}{\mathrm{/}}U = 0.50\). The other intersite Coulomb interactions are set as \((V_{b_2},V_p,V_q) = (0.56,0.66,0.58)\,V_{b_1}\), assuming the 1/rdependence. At n_{hole} = 1/2 (undoped case)^{29}, a firstorder phase transition occurs, with increasing \(U/t_{b_1}\), from a paramagnetic metal (PM) to a dimertype AF (DAF) phase in which the spins between dimers order in a staggered way as shown in Fig. 2b. This transition corresponds to the Mott MI transition. As \(U/t_{b_1}\) increases further, there appears a polar chargeordered (PCO) phase breaking the inversion symmetry^{31,34} with AF spin order, which can avoid the energy loss of \(V_{b_1}\), \(V_{b_2}\), and V_{p}, at the expense of the energy loss of V_{q} as shown schematically in Fig. 2c. The DAF and PCO phases are insulating at n_{hole} = 1/2. They have also been found in previous studies for the 3/4filled Hubbard models^{11,31,49} and the effective strong coupling models^{33,34}, and are stabilized in the relevant parameter regions for κ(ET)_{2}X. Experimentally, the DAF phase is widely observed in κ(ET)_{2}X as AF dimerMott insulator and the PCO phase is proposed to be related to the dielectric anomaly observed in κ(ET)_{2}Cu_{2}(CN)_{3} (κCN)^{36} and the insulating phase in κ(ET)_{2}Hg(SCN)_{2}Cl^{55}. Note that SC is a metastable state for \(U/t_{b_1}\) = 7–11.5, lying on each side of the DAFPCO boundary.
Away from n_{hole} = 1/2, significant doping asymmetry is observed and several different phases appear. For the holedoped side (n_{hole} > 1/2), while the PCO phase is rapidly suppressed, the DAF phase is enhanced to a smaller \(U/t_{b_1}\) region toward n_{hole} = 2/3. Note that in these phases the system becomes metallic once the doping is finite. Furthermore, a 3fold chargeordered (3fold CO1) phase appears for larger \(U/t_{b_1}\). The 3fold CO1 phase shows charge disproportionation and magnetic order as shown in Fig. 2d; holerich sites form a twodimensional network with AF spin order. This phase is insulating at n_{hole} = 2/3 (along the brown line in Fig. 2a) since the electron density fits the commensurability, and metallic for other hole densities because the excess holes can move through the ordered holes.
For the electrondoped side (n_{hole} < 1/2), the situation is much different. The PCO and DAF (both become metallic) phases are rapidly suppressed and another CO (3fold CO2) phase and a SC phase appear. The pattern of the charge disproportionation is opposite to that in the 3fold CO1 phase (hole rich ↔ hole poor) as shown in Fig. 1e; this configuration can fully avoid the intersite Coulomb interactions. Similar to the 3fold CO1 phase, the 3fold CO2 phase is insulating at n_{hole} = 1/3 (along the blue line in Fig. 2a) and metallic for other hole densities. Note that the 3fold CO2 phase is stabilized also for n_{hole} = 1/2 when \(V_{b_1}/U\) is larger (≥0.55)^{29,31} and is smoothly connected in the parameter space. While the 3fold CO1 phase is accompanied by the magnetic order, 3fold CO2 is nonmagnetic. We have tried several magnetic ordering patterns that coexist with 3fold CO2. However, none of them are stabilized because the CO pattern in the holerich sites forms a triangularlike structure and the spin degree of freedom is fully frustrated, and furthermore the distances between holerich sites are much longer than the original intermolecular bonds and therefore the effective magnetic exchange couplings are quite small. For smaller \(U/t_{b_1}\), the SC phase is realized by doping, located between the DAF/3fold CO2 and PM phases. The symmetry of the SC is the extendeds + \(d_{x^2  y^2}\)wave type, same with the one shown in our previous study at n_{hole} = 1/2^{29}. Details are discussed later.
Although SC does not appear as the ground state in the holedoped side, we find finite superconducting condensation energy in the phase diagram. Figure 2f shows the region where the condensation energy is finite, ignoring other ordered phases by setting Weiss fields to be zero in Φ〉. It is possible that the hidden SC phase appears if the DAF phase is destabilized by, e.g., disorder effect associated with doping or phase separation. Therefore, it is worthwhile to study the most favored SC phase even if it is a metastable state. While the d_{xy}wave SC is dominant for most of the holedoped side, the extendeds + \(d_{x^2  y^2}\)wave SC is stabilized for the electrondoped side. Namely, the symmetry of SC changes with carrier doping. Note that the charge correlation is greatly enhanced toward regions indicated by gray shade in Fig. 2f. In these regions, the mobility of holes are strongly restricted due to the strong intersite Coulomb interactions, and stable VMC simulations are difficult unless additional Weiss fields that induce longrange CO are introduced in Φ〉.
Fermi surface and spin structure factor
The electronhole doping asymmetry is closely related to the shape of the FS and the interdimer magnetic fluctuations. Figure 3a–c show the noninteracting FS for n_{hole} = 2/3, 1/2, and 1/3. As n_{hole} increases from 1/2 (hole doping), the FS shifts toward the right and left edges of the first Brillouin zone (MX line in Fig. 3b). Since the energy gap of the DAF order opens along the Brillouin zone edge^{11}, the DAF order becomes more favored for hole doping. For n_{hole} = 2/3, the FS almost touches the Brillouin zone edge, and there the DAF region extends down to \(U{\mathrm{/}}t_{b_1}\sim 2.5\), as shown in Fig. 2a. Note that the Fermi energy is located in the vicinity of van Hove singularity at n_{hole} = 2/3 as shown in Fig. 1b. Around this hole density, anomalous behavior such as pseudogap phenomena is naively expected^{46}. In clear contrast, the FS departs from the MX line for electron doping, consistent with the tendency of the DAF order being rapidly suppressed for n_{hole} < 1/2.
Next, Fig. 3d–f show the interdimer spin structure factor defined as,
for n_{hole} = 2/3, 1/2, and 1/3. Here, N_{dim}(=L^{2}) is the total number of dimers and \(M_l^{{\mathrm{dim}}} = (n_{2l  1 \uparrow } + n_{2l \uparrow })  (n_{2l  1 \downarrow } + n_{2l \downarrow })\) is the total spin density within lth dimer formed by molecular sites 2l − 1 and 2l with the central position r_{l}^{56}. Since the dimer centers form the anisotropic triangular lattice^{11}, the corresponding first Brillouin zone is the anisotropic hexagon. As shown in Fig. 3d, S^{dim}(q) for n_{hole} = 2/3 peaks around (0, ±π/R_{y}), which corresponds to the DAF spin configuration, suggesting that the AF spin fluctuation is enhanced by the Coulomb interactions and thus the DAF order is favored. On the other hand, for n_{hole} = 1/2, the peaks appear around six vertices of the Brillouin zone, as shown in Fig. 3e. This implies that the spin structure becomes more triangularlattice like (frustrated) and the AF spin fluctuation is suppressed as compared with that at n_{hole} = 2/3. For n_{hole} = 1/3, the peak structures almost diminish (see Fig. 3f), and the DAF order is not stabilized around this hole density.
Superconducting gap functions
The above mentioned electronhole asymmetry in the spin and the charge degrees of freedoms are the keys to understand the competition between the two types of SC. The main contribution of the gap function for the d_{xy}wave SC is given as
where α(=1, 2) denotes a band index, and \({\mathrm{\Delta }}_i^\alpha\) is the pairing with the ith neighbor dimers in the real space and treated as a variational parameter. We optimize the real space pairing up to 22nd neighbor dimers and find that the overall feature of the gap function is determined within the fourth neighbor, i.e., \({\mathrm{\Delta }}_m^\alpha\) for m ≤ 4. The term contaning \({\mathrm{\Delta }}_1^\alpha\) in Eq. (3) gives nodes in the horizontal (along k_{x}axis) and vertical (along k_{y}axis) directions. This is because the two diagonal pairings (orange and blue bars in Fig. 4a) have different sign, giving a d_{xy}type contribution. Therefore, this gap symmetry is referred to as d_{xy}wave^{26}. Note that the terms corresponding to real space pairings parallel to horizontal (along xaxis) and vertical (along yaxis) directions vanish since they are along the nodal directions. This SC phase has been discussed in analogy with that of highT_{c} cuprates, ascribing the diagonal directions in κtype structure to an approximate square lattice^{10}. As in the case for highT_{c} cuprates, the sign of the gap function on the FS changes four times (see Fig. 4c).
On the other hand, the main contribution of gap function for the extendeds+\(d_{x^2  y^2}\)wave SC is given as
The first term contaning \({\mathrm{\Delta }}_1^\alpha\) in Eq. (4) (blue bars in Fig. 4b) does not change sign within the first Brilloin zone and becomes zero only along the zone boundary; this term gives an extendedslike contribution. Furthermore, \({\mathrm{\Delta }}_1^\alpha\) changes sign between different bands, namely, \(sgn{\mathrm{\Delta }}_1^1 =  sgn{\mathrm{\Delta }}_1^2\). In this respect, the pairing symmetry can also be referred to as s_{±}, similar to that of ironbased SC^{57,58}. The second and third terms containing \({\mathrm{\Delta }}_2^\alpha\) and \({\mathrm{\Delta }}_3^\alpha\) in Eq. (4) (orange and yellow bars in Fig. 4b, respectively) give nodes in the diagonal direction; these terms give a \(d_{x^2  y^2}\)like contribution. The gap symmetry is thus referred to as an extendeds + \(d_{x^2  y^2}\)wave^{26,28,32,59}. The sign changes of the gap function on the FS is shown in Fig. 4d.
The competition between the two types of SC has been already discussed for the undoped case. First, in the effective 1/2filled dimer model, many early studies inferred the d_{xy}wave type^{12,13,14,16,17,18}. However, its stability over the AF phase has recently been doubted in several numerical studies including the VMC approach^{20,22}. Kuroki et al.^{26} were the first to point out the importance of treating the 3/4filled model, namely, considering the intradimer charge degree of freedom. In the 3/4filled Hubbard model, the two types of SC compete and either of them is favored depending on the parameters, especially the degree of dimerization^{26,28,32}. It is only recently that the importance of the intersite Coulomb interaction V_{ij} on SC was pointed out^{27,29}. In fact, our previous VMC study for the undoped case found that, although both symmetries are enhanced by the intersite Coulomb interactions, the extendeds + \(d_{x^2  y^2}\)wave is slightly more favored^{29}.
As shown in Fig. 2f, our calculations find that this competition is released when the mobile carriers are doped into the system. For the holedoped side, the AF spin fluctuation toward the DAF order is enhanced due to the shape of the FS and the Coulomb interactions, as already seen in Fig. 3, and consistent with the recent calculations based on the dimer model^{46}. Similarly to the highT_{c} cuprates, the AF spin fluctuation mediated SC is then developed with the strong singlet correlation along diagonal bonds shown in Fig. 4a, resulting in the d_{xy}wave symmetry. On the other hand, the AF spin fluctuation is suppressed with electron doping as seen in Fig. 3f and the spin singlet correlation along horizontal and vertical bonds compete with that along diagonal bonds. The extendeds + \(d_{x^2  y^2}\)wave is eventually favored since all bonds can contribute to the singlet pairing. The carrier doping deforms the shape of the FS and modifies the AF spin fluctuation, thus inducing the change of the symmetry of SC. Although the electronhole asymmetry shown in Fig. 2a appears similar to that of highT_{c} cuprates at a glance, they are much different especially for the SC phase. In both cases, the van Hove singularity appears only in the holedoped side, causing the electronhole asymmetry. However, different physics are delicately involved in κ(ET)_{2}X, as we have shown so far, and even the change of the symmetry of SC occurs by carrier doping. This is due to the unique geometrical frustration inherent in the triangularlike lattice structure of κ(ET)_{2}X. Furthermore, this asymmetry is expected to be robust for κ(ET)_{2}X in general because the van Hove singularity is always located in the holedoped side for the realistic parameter set of these conductors.
We can show more directly how the spin and charge correlations are correlated to the stability of the SC phases. The interdimer charge structure factor is defined as
where \(n_l^{{\mathrm{dim}}} = (n_{2l  1 \uparrow } + n_{2l \uparrow }) + (n_{2l  1 \downarrow } + n_{2l \downarrow })\) is the total charge density within lth dimer formed by molecular sites 2l − 1 and 2l with the central position r_{l}^{56}. Figure 5a–d show S^{dim}(q) and N^{dim}(q) for n_{hole} = 640/1152 = 0.556 and 544/1152 = 0.472, where the d_{xy}wave and the extendeds + \(d_{x^2  y^2}\)wave SC are stabilized, respectively. For n_{hole} = 0.556, S^{dim}(q) peaks around (0, ±π/R_{y}) that are favorable for the d_{xy}wave SC, while for n_{hole} = 0.472, S^{dim}(q) shows frustrated spin structure that are favorable for the extendeds + \(d_{x^2  y^2}\)wave SC. These are consistent with the mechanism of SC described above. On the other hand, N^{dim}(q) peaks around (0, ±3π/2R_{y}) for both n_{hole} = 0.556 and 0.472. This is because they are located near the instability of 3fold CO1 and 2, respectively, and the corresponding wave vectors are the same.
The superconducting condensation energy ΔE and spin/charge correlations are contrasting between the two SC phases. Figure 5e, f show the \(U/t_{b_1}\) dependence of N_{3fold}, S_{peak}, and ΔE. N_{3fold} = N^{dim}(0, ±3π/2R_{y}) and S_{peak} = S^{dim}(0, ±π/R_{y}) for n_{hole} = 0.556. The peak position in S^{dim}(q) changes along the upper and lower edges of the Brillouin zone for n_{hole} = 0.472 and we take the maximum value for S_{peak}. For n_{hole} = 0.556, ΔE does enhance, but despite the more rapid increase of N_{3fold} toward 3fold CO1 instability, it rather follows S_{peak}. This is consistent with the usual AF spin fluctuation picture discussed in highT_{c} cuprates. On the other hand, for n_{hole} = 0.472, ΔE is greatly enhanced following the rapid increase of N_{3fold}, which suggests the close correlation between the extendeds + \(d_{x^2  y^2}\)wave SC and the 3fold CO2 type charge fluctuation.
Discussion
Let us note that the intersite Coulomb interactions V_{ij} are indispensable to the stability of SC, not only for the extendeds + \(d_{x^2  y^2}\)wave SC phase but also for the metastable d_{xy}wave SC phase. As we have shown in the previous work^{29}, no longrange ordered phases are stabilized in the absence of V_{ij} for the undoped condition (or unphysically large U is necessary). This is also the case in the doped condition studied here. This indicates that in κ(ET)_{2}X, the charge degree of freedom is still active even with large U and the cooperation between U and V_{ij} induces various phases such as the DAF, PCO, 3fold COs, and SC. Especially, the extendeds + \(d_{x^2  y^2}\)wave SC is enhanced toward both polar and 3fold type CO instabilities. Recent experiment on photoinduced phase transition in κBr suggests that the polar charge oscillation is enhanced near the superconducting transition^{60}, consistent with our picture that the SC is enhanced toward the CO instability.
Experimentally, the symmetry of the SC for the undoped case is still controversial^{38,61,62,63,64,65,66,67}. This is consistent with our result suggesting that the competition between the two types of SC is most pronounced near the undoped n_{hole} = 1/2. Slight modification of parameters may alter the stability of the two. Now in κCl, both electron and hole doping are realized using the EDLT^{46,47}. The fillingcontrol MI transition and the emergence of SC have been confirmed with significant electronhole doping asymmetry. Our result suggests that the different SC appears by carrier doping; the d_{xy}wave SC on the holedoped side and the extendeds + \(d_{x^2  y^2}\)wave SC on the electrondoped side. The former is due to the strong AF spin fluctuation, similar to highT_{c} cuprates, and the latter is the consequence of frustrated spin structure and enhanced charge correlation under the geometrical frustration characteristic of the κtype ET compounds. Experiments for the chemically doped κ(ET)_{4}Hg_{3−δ}Y_{8} suggest similarities with the highT_{c} cuprates, especially for the nonFermiliquid behaviors above the SC transition temperature^{42,43,44}. This is overall consistent with our results because this system is holedoped. However, the tightbinding parameters for this system are rather closer to the isotropic triangularlattice like arrangement of dimers, where the AF phase is expected to be unfavored because of the stronger geometrical frustration. Indeed, the temperature dependence of the spin susceptibility is similar to that of κCN^{45}, which does not show any magnetic longrange orders. It is difficult to fully explain the character of κ(ET)_{4}Hg_{3−δ}Y_{8} within the present study and the analysis with appropriate tightbinding parameters is necessary for further understanding.
The nonmagnetic (candidate of gapless spinliquid) insulator and neighboring SC observed in κCN are also an intriguing phenomena for the undoped case. Although the tightbinding parameters and the resulting electronic structure of κCN is different from the present study, the intradimer charge degree of freedom and the intersite Coulomb interactions should play crucial roles as pointed out previously^{29}. Indeed, the anomalous dielectric response^{36} and Raman spectroscopy^{37} indicate that the intradimer charge degree of freedom is active in κCN. The stability of spinliquid and SC phase in the 1/2filled Hubbard model on the anisotropic triangular lattice, which is the approximate dimer model of κCN, is still controversial despite the long and extensive studies^{12,13,14,15,17,19,20,21,22,23,24,25}. The analysis for the 3/4filled EHM employed in this study will be an alternative way to investigate this issue and it is left for future studies.
Methods
Details of the VMC method
Here, we show the details of the VMC method. Trial wave function is a GutzwillerJastrow type, \(\left {\mathrm{\Psi }} \right\rangle = P_{{\mathrm{J}}_{\mathrm{c}}}P_{{\mathrm{J}}_{\mathrm{s}}}\left {\mathrm{\Phi }} \right\rangle\). Φ> is a onebody part constructed by diagonalizing the onebody Hamiltonian including the offdiagonal elements {D}, {M}, and {Δ} to treat longrange orders of charge, spin, and SC, respectively. The renormalized hopping integrals (\(\tilde t_{b_1},\tilde t_{b_2},\tilde t_p,\tilde t_q\)) are also included in Φ> as variational parameters, where \(\tilde t_{b_1} = t_{b_1}\) is fixed as a unit. \(P_{{\mathrm{J}}_{\mathrm{c}}} = {\mathrm{exp}}\,[  \mathop {\sum}\nolimits_{i,j} \,v_{ij}^{\mathrm{c}}n_in_j]\) and \(P_{{\mathrm{J}}_{\mathrm{s}}} = {\mathrm{exp}}\,[  \mathop {\sum}\nolimits_{i,j} \,v_{ij}^{\mathrm{s}}s_i^zs_j^z]\) are charge and spin Jastrow factors which control longrange charge and spin correlations, respectively. Here, \(s_i^z = n_{i \uparrow }  n_{i \downarrow }\), and we assume \(v_{ij}^{\mathrm{c}} = v^{\mathrm{c}}(r_i  r_j)\) and \(v_{ij}^{\mathrm{s}} = v^{\mathrm{s}}(r_i  r_j)\), where r_{i} is the position of molecular site i. The variational parameters in Ψ> are \(\tilde t_{b_2}\), \(\tilde t_p\), \(\tilde t_q\), {D}, {M}, \(\{ \Delta \}\), \(\{ v_{ij}^{\mathrm{c}}\}\), and \(\{ v_{ij}^{\mathrm{s}}\}\), and they are simultaneously optimized using the stochastic reconfiguration method^{68}. The total number of molecular sites is 4 × L × L/2 = 2L^{2} and varied from L = 12 to L = 24 with antiperiodic boundary conditions in both x and y directions of the primitive lattice vectors (see Fig. 1a).
The onebody part Φ> for the PM, DAF, and PCO states can be obtained by diagonalizing the onebody Hamiltonian,
with the hopping matrix elements given as
where \(c_{{\bf{k}}m\sigma }^\dagger\) (c_{kmσ}) is a creation (annihilation) operator of electron at molecule m(=1–4), as indicated in Fig. 1a, with momentum k and spin σ(=↑,↓), and s_{σ} = 1(−1) for σ = ↑(↓). \(M_m^z\) is a variational parameter which induces the staggered AF longrange order aligned to the z direction for molecule m; \(M_1^z = M_2^z = M_3^z = M_4^z = 0\) for the PM state, \(M_1^z = M_2^z = M_3^z = M_4^z \ne 0\) for the DAF state, and \(M_1^z = M_3^z \ne M_2^z = M_4^z\) for the PCO state. \(a_{{\bf{k}}\alpha \sigma }^\dagger\) (a_{kασ}) in Eq. (7) is a creation (annihilation) operator of quasiparticle in band α with momentum k and spin σ(=↑,↓), obtained by diagonalizing Eq. (6), and \(\tilde E_\alpha (\mathbf{k})\) is the corresponding quasipaticle energy.
For the 3fold CO1 and 2 states, the unit cell is three times larger than the original one and contains 12 molecules, as shown in Fig. 2d, e. Therefore, Φ> is obtained by diagonalizing a 12 × 12 matrix. The corresponding Weiss field is \(D_{m\sigma }\mathop {\sum}\nolimits_{{\bf{k}}\sigma } \,c_{{\bf{k}}m\sigma }^\dagger c_{{\bf{k}}m\sigma }\), which induces charge disproportionation and spin ordering within the unit cell through the variational parameters D_{mσ} for m = 1, 2, …, 12.
Finally, Φ> for SC is obtained by diagonalizing the BCStype meanfield Hamiltonian,
where Δ^{α} denotes gap function for band α and \(\xi _{\alpha} = \tilde E_{\alpha} ({\mathbf{k}})  {\tilde {\mu}}\) is a quasiparticle energy measured from the renormalized chemical potential \(\tilde \mu\). \({\mathrm{\Delta }}^\alpha = {\mathrm{\Delta }}^\alpha (\{ {\mathrm{\Delta }}_i^\alpha \} )\) is constructed from the real space pairing \({\mathrm{\Delta }}_i^\alpha\) up to the 22nd neighbor dimers (i = 1, 2, …, 22). Since the band 3 and 4 are located much below the Fermi energy (see Fig. 1b), their contribution to the pairing is negligible and therefore we set Δ^{3} = Δ^{4} = 0.
Data availability
The data that support the findings of this study are available from the corresponding author on reasonable request.
Code availability
The code that support the findings of this study are available from the corresponding author on reasonable request.
References
 1.
Bednorz, J. G. & Muller, K. A. Possible highT _{c} superconductivity in the BaLaCuO system. Z. Phys. B 64, 189 (1986).
 2.
Imada, M., Fujimori, A. & Tokura, Y. Metalinsulator transitions. Rev. Mod. Phys. 70, 1039 (1998).
 3.
Uchida, S. High Temperature Superconductivity, The Role to Higher Critical Temperature (Springer, Berlin, 2015).
 4.
Uemura, Y. J. et al. Basic similarities among cuprate, bismuthate, organic, Chevrelphase, and heavyfermion superconductors shown by penetrationdepth measurements. Phys. Rev. Lett. 66, 2665 (1991).
 5.
Sigrist, M. & Ueda, K. Phenomenological theory of unconventional superconductivity. Rev. Mod. Phys. 63, 239 (1991).
 6.
Matsuda, Y., Izawa, K. & Vekhter, I. Nodal structure of unconventional superconductors probed by angle resolved thermal transport measurements. J. Phys. Condens. Matter 18, R705 (2006).
 7.
Taillefer, L. Scattering and pairing in cuprate superconductors. Annu. Rev. Condens. Matter Phys. 1, 51 (2010).
 8.
Ardavan, A. et al. Recent topics of organic superconductors. J. Phys. Soc. Jpn. 81, 011004 (2012).
 9.
Lebed, A. The Physics of Organic Superconductors and Conductors (Springer, New York, 2008).
 10.
McKenzie, R. H. Similarities between organic and cuprate superconductors. Science 278, 820 (1997).
 11.
Kino, H. & Fukuyama, H. Phase diagram of twodimensional organic conductors: (BEDTTTF)_{2}X. J. Phys. Soc. Jpn. 65, 2158 (1996).
 12.
Kino, H. & Kontani, H. Phase diagram of superconductivity on the anisotropic triangular lattice Hubbard model: An effective model of κ(BEDTTTF) salts. J. Phys. Soc. Jpn. 67, 3691 (1998).
 13.
Kondo, H. & Moriya, T. Spin fluctuationinduced superconductivity in organic compounds. J. Phys. Soc. Jpn. 67, 3695 (1998).
 14.
Schmalian, J. Pairing due to spin fluctuations in layered organic superconductors. Phys. Rev. Lett. 81, 4232 (1998).
 15.
Morita, H., Watanabe, S. & Imada, M. Nonmagnetic insulating states near the Mott transitions on lattices with geometrical frustration and implications for κ(ET)_{2}Cu_{2}(CN)_{3}. J. Phys. Soc. Jpn. 71, 2109 (2002).
 16.
Liu, J., Schmalian, J. & Trivedi, N. Pairing and superconductivity driven by strong quasiparticle renormalization in twodimensional organic charge transfer salts. Phys. Rev. Lett. 94, 127003 (2005).
 17.
Kyung, B. & Tremblay, A.M. S. Mott transition, antiferromagnetism, and dWave superconductivity in twodimensional organic conductors. Phys. Rev. Lett. 97, 046402 (2006).
 18.
Sahebsara, P. & Sénéchal, D. Antiferromagnetism and superconductivity in layered organic conductors: variational cluster approach. Phys. Rev. Lett. 97, 257004 (2006).
 19.
Koretsune, T., Motome, Y. & Furusaki, A. Exact diagonalization study of Mott transition in the Hubbard model on an anisotropic triangular lattice. J. Phys. Soc. Jpn. 76, 074719 (2007).
 20.
Watanabe, T., Yokoyama, H., Tanaka, Y. & Inoue, J. Predominant magnetic states in the Hubbard model on anisotropic triangular lattices. Phys. Rev. B 77, 214505 (2008).
 21.
Shinaoka, H., Misawa, T., Nakamura, K. & Imada, M. Mott transition and phase diagram of κ(BEDTTTF)_{2}Cu(NCS)_{2} studied by twodimensional model derived from ab initio method. J. Phys. Soc. Jpn. 81, 034701 (2012).
 22.
Dayal, S., Clay, R. T. & Mazumdar, S. Absence of longrange superconducting correlations in the frustrated halffilledband Hubbard model. Phys. Rev. B 85, 165141 (2012).
 23.
Tocchio, L. F., Feldner, H., Becca, F., Valentí, R. & Gros, C. Spinliquid versus spiralorder phases in the anisotropic triangular lattice. Phys. Rev. B 87, 035143 (2013).
 24.
Laubach, M., Thomale, R., Platt, C., Hanke, W. & Li, G. Phase diagram of the Hubbard model on the anisotropic triangular lattice. Phys. Rev. B 91, 245125 (2015).
 25.
Shirakawa, T., Tohyama, T., Kokalj, J., Sota, S. & Yunoki, S. Groundstate phase diagram of the triangular lattice Hubbard model by the densitymatrix renormalization group method. Phys. Rev. B 96, 205130 (2017).
 26.
Kuroki, K., Kimura, T., Arita, R., Tanaka, Y. & Matsuda, Y. d_{x} ^{2} _{−y} ^{2}  versus d_{xy} like pairings in organic superconductors κ(BEDTTTF)_{2}X. Phys. Rev. B 65, 100516(R) (2002).
 27.
Sekine, A., Nasu, J. & Ishihara, S. Polar charge fluctuation and superconductivity in organic conductors. Phys. Rev. B 87, 085133 (2013).
 28.
Guterding, D., Altmeyer, M., Jeschke, H. O. & Valentí, R. Neardegeneracy of extended s + d_{x} ^{2} _{−} _{y} ^{2} and d_{xy} order parameters in quasitwodimensional organic superconductors. Phys. Rev. B 94, 024515 (2016).
 29.
Watanabe, H., Seo, H. & Yunoki, S. Phase competition and superconductivity in κ(BEDTTTF)_{2}X: Importance of intermolecular Coulomb interactions. J. Phys. Soc. Jpn. 86, 033703 (2017).
 30.
Powell, B. J., Kenny, E. P. & Merino, J. Dynamical reduction of the dimensionality of exchange interactions and the “spinLiquid” phase of κ(BEDTTTF)_{2}X. Phys. Rev. Lett. 119, 087204 (2017).
 31.
Kaneko, R., Tocchio, L. F., Valentí, R. & Becca, F. Charge orders in organic chargetransfer salts. New J. Phys. 19, 103033 (2017).
 32.
Zantout, K., Altmeyer, M., Backes, S. & Valentí, R. Superconductivity in correlated BEDTTTF molecular conductors: critical temperatures and gap symmetries. Phys. Rev. B 97, 014530 (2018).
 33.
Hotta, C. Quantum electric dipoles in spinliquid dimer Mott insulator κET_{2}Cu_{2}(CN)_{3}. Phys. Rev. B 82, 241104(R) (2010).
 34.
Naka, M. & Ishihara, S. Electronic ferroelectricity in a dimer Mott insulator. J. Phys. Soc. Jpn. 79, 063707 (2010).
 35.
Manna, R. S., de Souza, M., Brühl, A., Schlueter, J. A. & Lang, M. Lattice effects and entropy release at the lowtemperature phase transition in the spinliquid candidate κ(BEDTTTF)_{2}Cu_{2}(CN)_{3}. Phys. Rev. Lett. 104, 016403 (2010).
 36.
AbdelJawad, M. et al. Anomalous dielectric response in the dimer Mott insulator κ(BEDTTTF)_{2}Cu_{2}(CN)_{3}. Phys. Rev. B 82, 125119 (2010).
 37.
Yakushi, K., Yamamoto, K., Yamamoto, T., Saito, Y. & Kawamoto, A. Raman spectroscopy study of charge fluctuation in the spinliquid candidate κ(BEDTTTF)_{2}Cu_{2}(CN)_{3}. J. Phys. Soc. Jpn. 84, 084711 (2015).
 38.
Guterding, D. et al. Evidence for eightnode mixedsymmetry superconductivity in a correlated organic metal. Phys. Rev. Lett. 116, 237001 (2016).
 39.
Kagawa, F., Miyagawa, K. & Kanoda, K. Unconventional critical behaviour in a quasitwodimensional organic conductor. Nature 436, 534 (2005).
 40.
Kurosaki, Y., Shimizu, Y., Miyagawa, K., Kanoda, K. & Saito, G. Mott transition from a spin liquid to a fermi liquid in the spinfrustrated organic conductor κ(ET)_{2}Cu_{2}(CN)_{3}. Phys. Rev. Lett. 95, 177001 (2005).
 41.
Kanoda, K. MetalInsulator transition in κ(ET)_{2}X and (DCNQI)_{2}M: two contrasting manifestation of electron correlation. J. Phys. Soc. Jpn. 75, 051007 (2006).
 42.
Naito, A. et al. Anomalous enhancement of electronic heat capacity in the organic conductors κ(BEDTTTF)_{4}Hg_{3−δ}X_{8} (X = Br, Cl). Phys. Rev. B 71, 054514 (2005).
 43.
Taniguchi, H. et al. Anomalous pressure dependence of superconductivity in layered organic conductor, κ(BEDTTTF)_{4}Hg_{2.89}Br_{8}. J. Phys. Soc. Jpn. 76, 113709 (2007).
 44.
Oike, H., Miyagawa, K., Taniguchi, H. & Kanoda, K. Pressureinduced Mott transition in an organic superconductor with a finite doping level. Phys. Rev. Lett. 114, 067002 (2015).
 45.
Oike, H. et al. Anomalous metallic behaviour in the doped spin liquid candidate κ(BEDTTTF)_{4}Hg_{2.89}Br_{8}. Nat. Commun. 8, 756 (2017).
 46.
Kawasugi, Y. et al. Electronhole doping asymmetry of Fermi surface reconstructed in a simple Mott insulator. Nat. Commun. 7, 12356 (2016).
 47.
Sato, Y., Kawasugi, Y., Suda, M., Yamamoto, H. M. & Kato, R. Critical behavior in dopingdriven metalinsulator transition on singlecrystalline organic MottFET. Nano Lett. 17, 708 (2017).
 48.
Seo, H., Hotta, C. & Fukuyama, H. Toward systematic understanding of diversity of electronic properties in lowdimensional molecular solids. Chem. Rev. 104, 5005 (2004).
 49.
Seo, H. Charge ordering in organic ET compounds. J. Phys. Soc. Jpn. 69, 805 (2000).
 50.
Miyagawa, K., Kawamoto, A. & Kanoda, K. Proximity of pseudogapped superconductor and commensurate antiferromagnet in a quasitwodimensional organic system. Phys. Rev. Lett. 89, 017003 (2002).
 51.
Koretsune, T. & Hotta, C. Evaluating model parameters of the κ and β′type Mott insulating organic solids. Phys. Rev. B 89, 045102 (2014).
 52.
McMillan, W. L. Ground state of liquid He^{4}. Phys. Rev. 138, A442 (1965).
 53.
Ceperley, D., Chester, G. V. & Kalos, M. H. Monte Carlo simulation of a manyfermion study. Phys. Rev. B 16, 3081 (1977).
 54.
Yokoyama, H. & Shiba, H. Variational MonteCarlo studies of Hubbard Model. I. J. Phys. Soc. Jpn. 56, 1490 (1987).
 55.
Drichko, N. et al. Metallic state and chargeorder metalinsulator transition in the quasitwodimensional conductor κ(BEDTTTF)_{2}Hg(SCN)_{2}Cl. Phys. Rev. B 89, 075133 (2014).
 56.
Yoshimi, K., Seo, H., Ishibashi, S. & Brown, S. E. Tuning the magnetic dimensionality by charge ordering in the molecular TMTTF salts. Phys. Rev. Lett. 108, 096402 (2012).
 57.
Mazin, I. I., Singh, D. J., Johannes, M. D. & Du, M. H. Unconventional superconductivity with a sign reversal in the order parameter of LaFeAsO_{1−x}F_{x}. Phys. Rev. Lett. 101, 057003 (2008).
 58.
Kuroki, K. et al. Unconventional pairing originating from the disconnected fermi surfaces of superconducting LaFeAsO_{1−x}F_{x}. Phys. Rev. Lett. 101, 087004 (2008).
 59.
Powell, B. J. & McKenzie, R. H. Symmetry of the superconducting order parameter in frustrated systems determined by the spatial anisotropy of spin correlations. Phys. Rev. Lett. 98, 027005 (2007).
 60.
Kawakami, Y. et al. Nonlinear charge oscillation driven by a singlecycle light field in an organic superconductor. Nat. Photonics 12, 474 (2018).
 61.
Elsinger, H. et al. κ(BEDTTTF)_{2}Cu[N(CN)_{2}]Br: a fully gapped strongcoupling superconductor. Phys. Rev. Lett. 84, 6098 (2000).
 62.
Izawa, K., Yamaguchi, H., Sasaki, T. & Matsuda, Y. Superconducting gap structure of κ(BEDTTTF)_{2}Cu(NCS)_{2} probed by thermal conductivity tensor. Phys. Rev. Lett. 88, 027002 (2001).
 63.
Taylor, O. J., Carrington, A. & Schlueter, J. A. Specificheat measurements of the gap structure of the organic superconductors κ(ET)_{2}Cu[N(CN)_{2}]Br and κ(ET)_{2}Cu(NCS)_{2}. Phys. Rev. Lett. 99, 057001 (2007).
 64.
Ichimura, K., Takami, M. & Nomura, K. Direct observation of dwave superconducting gap in κ(BEDTTTF)_{2}Cu[N(CN)_{2}]Br with scanning tunneling microscopy. J. Phys. Soc. Jpn. 77, 114707 (2008).
 65.
Malone, L., Taylor, O. J., Schlueter, J. A. & Carrington, A. Location of gap nodes in the organic superconductorsκ(ET)_{2}Cu(NCS)_{2} and κ(ET)_{2}Cu[N(CN)_{2}]Br determined by magnetocalorimetry. Phys. Rev. B 82, 014522 (2010).
 66.
Milbradt, S. et al. Inplane supefluid density and microwave conductivity of the organic superconductor κ(BEDTTTF)_{2}Cu[N(CN)_{2}]Br: Evidence for dwave pairing and resilient quasiparticles. Phys. Rev. B 88, 064501 (2013).
 67.
Oka, Y. et al. Tunneling spectroscopy in organic superconductor κ(BEDTTTFd[3,3])_{2}Cu[N(CN)_{2}]Br. J. Phys. Soc. Jpn. 84, 064713 (2015).
 68.
Sorella, S. Generalized Lanczos algorithm for variational quantum Monte Carlo. Phys. Rev. B 64, 024512 (2001).
Acknowledgements
The authors thank R. Kato, Y. Kawasugi, H. Itoh, S. Iwai, Y. Kawakami, and M. Naka for useful discussions. The computation has been done using the facilities of the Supercomputer Center, Institute for Solid State Physics, University of Tokyo. This work has been supported by JSPS KAKENHI (Grant Nos 26800198, 26400377, 16H02393, and 18H01183) and Waseda University Grant for Special Research Projects (Project Number: 2018B352).
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H.W. performed the VMC simulations and prepared the figures. Results were analyzed and the paper was written by H.W., H.S., and S.Y.
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Watanabe, H., Seo, H. & Yunoki, S. Mechanism of superconductivity and electronhole doping asymmetry in κtype molecular conductors. Nat Commun 10, 3167 (2019). https://doi.org/10.1038/s41467019110221
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